6 Investigation

 
 

 

 

1  Rhombus Mosaics

In this exercise mosaics will be made out of rhombuses. According to the dictionary a rhombus is a quadrilateral with four equal sides.
So, according to our theorem in section 4, a rhombus is a parallelogram. Hence the opposite angles in a rhombus are equal and adjacent angles add up to 180 degrees.
The decagon in the picture is divided in red and yellow rhombuses. All red ones are congruent and so are the yellow ones. There are four different angles in the diagram.

a  Calculate a, b and c.
Write down your calculation as well.

In the applet rhombuses are used to build polygons. If you start with 5 rhombuses you get the decagon.

b  Use the applet to see what happens when
you vary the number of rhombuses. This it what
the following questions are about.

The number of rhombuses is called n. Your teacher has a sheet showing all polygons the applet provides for n = 5, ..., 13. It comes in handy when answering the next questions. Take n = 7. The shape consists of 21 rhombuses in three rings. In each ring all rhombuses are congruent.

c  Calculate the angles of every type of rhombus in the mosaic. Write down your calculations again.

The 14-sided polygon that emerges looks regular. Suppose it is regular, then all angles have to be equal.

d  Make a calculation to check if this is true.

Take n = 8 now. You may wonder if the polygon that appears has 8 vertices or 16.

e  Answer this question. Explain your answer.

f  Fill in the table.
n 5 6 7 8 9 10 11 12 13
number of rings in the mosaic                                                                        
a regular ......-sided polygon appears                                                                        
total number of rhombuses in the mosaic                                                                        
g  What relations can you find in the table?

h  What numbers would be in the table for n = 52 and n = 53?

There is a relation between the angles of the rhombuses in consecutive rings of the mosaic.

i  Find that relation.

Squares appear in the mosaic when you take n = 8.

j  For what other values of n does this happen? In which ring?
applet
 

 

 

2  Pentagonal tiles



Floors consisting of pentagon tiles are rare. I happened to see such a floor in the palace of the elector in Heidelberg (Germany). The tiles have been there for ages, so the edges were not exactly straight anymore, nor did they fit together. Suppose we were able to perfect the floor, then it would look like the tiling above.

The following is given:
  • all tiles are congruent
  • the tiles fit together exactly
  • every tile has two b° angles, two c° angles and one of a°
In the diagram you can see once more how three tiles are put together. Some vertices are shared by three tiles, some are shared by four tiles. Have a look at those points and you will find that
4b = 360 and that a + 2c = 360.

a  Explain this.

b  So what is b?

c  Use the angle sum of a pentagon to show that the value of a +2c is correct.

The four sides marked with * are of the same length. The length of the fifth side depends on the size of a.

d  Choose a = 120 and take 4 cm for the sides that have the same length. Draw the tile you will get this way as accurately as possible. Measure the length of the fifth side.

In the type of tiles we use a can never be 80°.

e  Explain why.

f  What values can a have?

The tiles in the Heidelberg floor have five equally long sides. This is only true for a unique value of a.

g  Draw such a tile; take 4 cm sides. Measure a accurately.

 
 

 

 

3  Rods and congruence

Part 1
Four rods have been connected with nails. It is not a very solid construction. When you push one of the rods, the construction will change shape easily.

In the applet you will see how that construction changes shape when you push it.
You can strengthen your construction by using an extra rod and two nails.






When you attach the extra rod between two adjacent sides (as shown in the diagram), the new construction will become rigid.

a  Explain why.

Suppose the rod quadrilateral is a parallelogram.
When you attach the extra rod between two opposite sides, the new construction doesn't have to be rigid.

b  How should you attach the rod for that to happen?

A rod pentagon is not solid at all of course.

c  How many extra rods do you need to make it rigid?

d  What about a rod octagon?
How many extra rods would you need?

Part 2
Two triangles in which corresponding sides are equal are congruent (SSS). We will study congruence of quadrilaterals now.

a  Draw three different quadrilaterals ABCD with AB = 3 cm, BC = 4 cm, CD = 5 cm and
DA = 6 cm.
Is SSSS one of the congruence theorems for quadrilaterals? In other words: Are two quadrilaterals in which all corresponding sides are equal congruent?

In quadrilateral ABCD the following are given: AB = 5 cm, B = 90°, BC = 6 cm, C = 83° and CD = 3 cm.

b  Draw such a quadrilateral ABCD.
Are there more possibilities, or are all such quadrilaterals congruent?
Is SASAS a congruence theorem for quadrilaterals?

In quadrilateral ABCD the following are given: AB = 3 cm, B = 90°, BC = 6 cm, CD = 3 cm and DA = 5 cm.

c  Draw such a quadrilateral ABCD.
Are there more possibilities, or are all such quadrilaterals congruent?
Is SASSS a congruence theorem for quadrilaterals?

In quadrilateral ABCD the following are given: A = 65°, AB = 5 cm, B= 90°, BC = 6 cm and C = 83°.

d  Draw such a quadrilateral ABCD.
Are there more possibilities, or are all such quadrilaterals congruent?
Is ASASA one of the congruence theorems for quadrilaterals?

e  Name a congruence theorem for pentagons.
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